NASA t'ontractor Report 191055

/

Geometrically Nonlinear Analysis ofLaminated Elastic Structures

J.N. Reddy, K. Chandrashekhara, and W.C. Chao

Virgima Polytechnic Institute and State University

Blacksburg, Virginia

January 1993

Prepared for

Lewis Research Center

Under Grant NAG3-208

N/ ANational Aeronautics and

Space Administration

(NABA-CR-I9!055)

NC_LINLA/_, ANALYSIS

LLAST IC STRUCTURES

Polytechnic Inst..

l:)Z p

_EUMETRICALLY

13F LAMINATFD

(Virginiaand State Univ.)

N93-17924

Unclas

G3124 0142844

Part 1:

Part 2:

Part 3:

Table of Contents

Geometrically Nonlinear Analysis of Laminated Shells Including

Transverse Shear Strains by J. N. Reddy and K. Chandrashekhara

Summary; Introduction; A Review of Governing Equations; Finite

Clement Model; Numerical Results; Conclusions; References;

Appendix.

Analysis of Laminated Composite Shells Using a Degenerated 3-D

Element by W. C. Chao and J. N. Reddy

Summary; Introduction; Incremental, Total-LagrangianFormulation of a Contiuous Medium; Finite Element Model;

Discussionofthe Numerical Results; References.

Nonlinear Material Models for Composite Plates and Shells by K.

Chandrashekhara and J. N. Reddy

Sunvnary; Introduction; Material Model; Finite Element

Formulation; Numerical Results; Conclusions; References.

ii

GEOMETRICALLYNONLINEARANALYSISOF LAMINATEDELASTICSTRUCTURES

J.N. Reddy, K. Chandrashekhara° and W.C. ChaoVirginia Polytechnic Institute and State University

Blacksburg, Virginia 24061

ABSTRACT

This final technical report contains three parts: Part i

deals with the 2-D shell theory and its element formulation and

applications. Part 2 deals with the 3-D degenerated element.

These two parts constitute the two major tasks that werecompleted under the grant. Another related topic that was

initiated during the present investigation is the development of

a nonlinear material model. This topic is briefly discussed in

Part 3. To make each part self-contained, conclusions and

references are included in each part. In the interest of

brevity, the discussions presented here are relatively brief.The details and additional topics are described in the referencescited.

PART 1

GEOMETRICALLY NONLINEAR ANALYSIS OF LAMINATED SHELLS

INCLUDING TRANSVERSE SHEAR STRAINS

J. N. Reddy and K. Chandrashekhara t

(A condensed uersion of t_ paper is to appear _ AIAA Jo_u_l, 1984)

SUMMARY

The paper contains a description of a doubly curved shell finite element

for geometrically nonlinear (in the yon Karman sense) analysis of laminated

(doubly-curved) composite shells. The element is based on an extension of the

Sanders snell theory and accounts for the von Karman strains and transverse

shear strains. The numerical accuracy and convergence characteristics of the

element are further evaluated by comparing the present results for the bending

of isotropic and orthotropic plates and shells with those available in the

literature. The many numerical results presented here for the geomertically

nonlinear analysis of laminated composite shells should serve as reference for

future investigations.

INTRODUCTION

Laminated shells are finding increased application in aerospace, automo-

bile and petrochemical industries. This is primarily due to the high stiff-

ness to weight ratio, high strength to weight ratio, and less machining and

?

maintenance costs associated with composite structures. However, the analysis

of composite structures is more complicated when compared to metallic struc-

tures, because laminated composite structures are anisotropic and character-

ized by bending-stretching coupling. Further, the classical snell theories,

which are based on the Kirchhoff-Love kinematic hypothesis (see Naghdi [i] and

Graduate research assistant

2

Bert [2]) are knownto yield deflections and stresses in _aminated shells that

are as muchas 30%in error. This error is due to the neglect of transverse

shear strains in the classical shell theories.

Refinements of the classical shell theories (e.g., Love's first approxi-

mation theory [3]) for shells to include transverse shear deformation have

been presented by Reissner [4-6]. Sanders [7] presented modified first- and

second-approximation theories that removedan inconsistency (nonvanishing of a

small rigid-body rotations of the shell) existed in Love's first-approximation

theory.

The first thin shell theory of laminated orthotropic composite shells is

due to Ambartsumyan[8,9]. In these works Ambartsumyanassumedthat the indi-

vidual ortbotropic layers were oriented such that the principal axes of mate-

rial symmetry coincided with the principal coordinates of the shell reference

surface. Dong, Pister, and Taylor [10] presented an extension of Donnell's

shallow snell theory [11] to thin laminated shells. Using the asymptotic in-

tegration of the elasticity equations, Widera and Chung[12] derived a first-

approximation theory for the unsymmetric deformation of nonbomogeneous,aniso-

tropic, cylindrical shells. This theory, when specialized to isotropic mate-

rials, reduces to Donnell's snell theory.

The effects of transverse shear deformation and thermal expansion through

the shell thickness were considered by Zukas and Vinson [13]. Dong and Tso

[14] constructed a laminated orthotropic shell theory that includes transverse

shear deformation. This theory can be regarded as an extension of Love's

first-approximation theory [3] for homogeneous isotropic shells. Other re-

fined theories, specialized to anisotropic cylindrical shells, were presented

by Whitney and Sun [15], and Widera and Logan [16,17].

The finite-element analysis of layered anisotropic shells, all of which

are concerned with bending, stability, or vibration of shells, can be found in

the works of Scnmit and Monforton [18], Panda and Natarajan [19], Sbivakumar

and Krishna Murty [20], Rao [21], Siede and Chang[22], Hsu, Reddy, and Bert

[23], Reddy[24], and Venkatesh and Rao [25,26]. Recently, Reddy [27] extend-

ed the Sanders theory to account for the transverse shear strains, and pre-

sented exact solutions for simply supported cross-ply laminated shells. All

of these studies are limited small displacement theories and static analyses.

In the present paper, an extension of the Sanders shell theory that ac-

counts for the shear deformation and the von Karmanstrains in laminated an-

isotropic shells is used to develop a displacement finite element model for

the bending analysis of laminated composite shells. The accuracy of the ele-

ment is evaluated by comparing the results obtained in the present study for

isotropic and orthotropic plate and shell problems with those available in the

literature. Numerical results for bending analysis of cylindrical and doubly-

curved smells are presented, showing the effect of radius-to-thickness ratio,

loading, and boundary conditions on the deflections and stresses.

A REVIEW OF THE GOVERNING EQUATIONS

Consider a laminated shell constructed of a finite number of uniform-

thickness orthotropic layers, oriented arbitrarily with respect to the shell

coordinates ({1,{2,_). The orthogonal curvilinear coordinate system

({i,{2,_) is chosen such that the {I- and {2" curves are lines of curvature on

the midsurface {=0, and _-curves are straight lines perpendicular to the sur-

face {=0 (see Fig. I). A line element of the shell is given by (see Reddy

[27]

(dS) 2 : [(i + _/Rl)_Id_l ]2 + [(I + _/R2)_2d_2 ]2 + (d_)2 (I)

4

J_

I_,.r

C_J

x

\\

_ur

x

Q;J_

"oQJ

c_

0

.pJ

E

i.m-

0

e-0

.r.-

0

e-

Q;

0

.._I--

5

where mi and Ri (i : 1,2) are the surface metrics and radii of curvature,

respectively. In general, _i and Ri are functions of _i only. For the doubly

curved shells considered in the present theory, _i and Ri are constant.

The strain-displacement equations of the shear deformable theory of

doubly-curved shells are given by

o¢i : el + _I

o

e2 = ¢2 + _2o

_4 = ¢4

oE5 = ¢5

o_6 = e6 + _6

where

_Ul u3 _u3 _*I

(2)

_u2 + u3 _u 3• ' _2

_)x2

o _Ul _)u2 _u3 _u3 9¢1 9¢2 _u2 _Ul

¢6 : _ + + , = + + c C_TIo

_u 3 u 2

_ : _b2 + _x 2 - R'-2"

o _)u3 Ul

¢5 : ¢I + _-_i - RT

I 1 I: - -- - --) (3)

Co 2 CR2 R1

Here ui denotes the displacements of the reference surface along _i(_ 3 = _)

axes, and _i are the rotations of the transverse normals to the reference

surface. In Love's first-approximation theories the parameter co is taken to

be zero, and it is introduced only in the Sanders theory.

6

The stress-strain relations, transformed to the shell coordinates, are of

the form

{_} : [q]{c} (4)

^(k)where t_ij are the material properties of k-in layer.

The principle of virtual work for the present problem is given by

L

k=1 _k-1 Q

- q6u3}_l_2d{id_zd_ (5)

+ Q26_ q6u3]=l=2d_Id_2 {6)

where q is the distributed transverse load, Ni and Mi are the stress and

moment resultants, and Qi is the shear force resultant:

L _k

(Ni,Mi): Z Sk=1 _K-I

L _k

Qi = Z K_ S _i d_ ,k:l _k=l

, i = 1,2,6

i = 4,5, (7)

where Ki (i : 1,2) are the shear correction factors (taken to be K_ : K_ :

5/6), and (_k.1,_k) are the _-coordinates of the k-th layer, and L is the

total number of layers in the laminated shell.

It is informative to note that the equations of equilibrium can be

derived from Eq. (6) by integrating the displacement gradients in _ by parts

and setting the coefficients of 6ui (i : 1,2,3) and 6¢i ('i:i,2) to zero1 1 1

separately. Weobtain [with co : _ (_- _) and dxi = (_id_i ]

_NI _ QI :+_x--7 _x2 (N6-tom6)+R-T 0

_N2 Q2

_x--[("6÷Co"6)÷_x--_+_= 0

aQ1 aQ2 N1 N2+ NCu3): o

M1 _ M6_+_-QI=0_x I _x 2

_M6 _M2

_+ _T2 " Q2 =0_)xI(8)

where

_ _ _u3 _u 3 _ _u3NCu31_xIC"i_x-_÷ "6_-T_2)÷ _x2 C"6_x--T (9)

The resultants (Ni, Mi, Qi ) are related to (¢_, (i) (i,j : 1,2,6) by

O+ BNi : Aij{ j ij_j

o + (I0)Mi = BijEj Dij(j

0 0

Q2 = A44¢4 + A45{5 (11)0 0

QI = A45% + A55¢5

T

Here Aij, Bij and Dij (i,j = 1,2,6) denote the extensional, flexural-

extensional coupling, and flexural stiffnesses of the laminate:

(Aij,Bij,Dij) :

L _k

k=1 _k-1

Ik) (l,_,_2)d_ (i,j : 1,2,6)Qj

(12)

8

(A44,A45,A55) = Z (K_Q44 ,KIK2Q45k:1 I k-1

The boundary conditions, derived froln the virtual work statement, involve

specifying either the essential boundary conditions (EBC) or the natural

boundary conditions (NBC):

EBC

uI or

u2 or

u3 or

NBC

Nln I + (N 6 - CoM6)n 2

N2n2 + (N 6 + CoM6)n2

{)u3 _u 3

(N1 + (N2 - z)n2_u3 _u3

+ (N6 _-_-2)nI + (N6 _-_-l)n2

+ Q2n2 + Q1nl

01 or Mln I + M6n2

02 or M2n2 + M6nI (13)

where (nl,n 2) denote the direction cosines of the unit normal on the boundary

of the midsurface of the shell.

T_e exact form of the spatial variation of the solution of Eqs. (8)-(13),

for the small-displacement theory, can be obtained under the following condi-

tions (see Reddy [27]):

(i) Symmetric or antisymmetric cross-ply laminates: i.e., laminates

with

A16 = A26 = B16 = B26 = D16 = D26 = A45 = O. (14)

(ii) Freely supported boundary conditions:

_i(0,x2) = _1(a,x2) : Ml(O,x 2) : Ml(a,x 2) : 0

u3(0,x2) : u3(a,x2) : u2(0,x2) : u2(a,x 2) : 0

9

N2(xI,O) : N2(Xl,b) : M2(xI,0) : M2(Xl,b) : 0

u3(xl,O) : u3(Xl,b) = Ul(Xl,O) : Ul(Xl,b) = 0

¢2(0,x2) : _2(a,x2) : _1(Xl,O) : ¢l(Xl,b) : 0 (15)

where a and b are the dimensions of the shell middle surface along

the xI and x2 axes, respectively. The time variation of the load

does not influence the spatial form of the solution.

Note that the exact solution can be obtained only for cross'ply laminated

shells with simply supported boundary conditions. For general lamination

schemes, exact solutions are not available to date.

FINITE-ELEMENT MODEL

A typical finite element is a doubly-curved shell element in the XlX 2-

surface. Over the typical shell element _(e), the displacements

(Ul,U2,U3,¢1,¢2) are interpolated by expressions of the form,

N

ui : jZ:I u_bj(x1'x2), i = 1,2,3

N

= J (xI,x2) , i = 1,2 (16)¢i _ ¢i*jj=1

J and Jwhere _j are the interpolation functions, and u i ¢i are the nodal values

of u i and ¢i' respectively. For a linear isoparametric element (N = 4) this

interpolation results in a stiffness matrix of order 20 by 20. For a nine-

node quadratic element the element stiffness matrix is of order 45 by 45.

Substitution of Eq. (21) into the virtual work principle, Eq. (9) yields

an element equation of t_e form

10

[K(_)] {a} : {F} (17)

where {4} : {{Ul}, {u2}, {u3}, {¢i}, {¢2}} T [K] the element stiffness matrix

and {F} is the force vector. In the interest of brevity, the coefficients of

the stiffness matrices are included in Appendix I.

The element equations (17) can be assembled, boundary conditions can be

imposed, and the resulting equations can be solved at each load step. Note

that the stiffness matrix [K] is a function of the unknown solution vector

{4}; therefore, an iterative solution procedure is required for each load

step. In the present study, we used the direct iteration technique, which can

be expressed as

[K({_}r)]{_} r+l : {F} (18)

where {_}r denotes the solution vector obtained in the r-th iteration (at any

given load step). At the beginning of the first load step, we assume that

{A}o : {0} and obtain the linear solution at the end of the first iteration.

The solution obtained at the end of the r-th iteration is used to compute the

stiffness matrix for the (r+l)-th iteration, At the end of each iteration

(for any load step), the solutions obtained in two consecutive iterations are

compared to see if they are close enough to terminate the iteration and to

move on to the next load step. The following convergence criterion is used in

the present study:

N .r+l N 211/2[ [ - 12/ o.oi (19)i:I i=1

where N is the total number of unknown generalized displacements in the finite

element mes_.

11

To accelerate the convergence, a weighted average o_ the solution from

last two iterations are used to compute the stiffness matrix:

[K(y{A} r'l + (I- y){A}r)]{a} r+l : {F} (20)

where y is the acceleration parameter, 0 < y < I. In the present study a val-

ue of 0.25 - 0.35 was used.

NUMERICAL RESULTS

Here we present numerical results for some sample problems. To illus-

trate the accuracy of the present element, first few examples are taken from

the literature on isotropic and orthotropic shells. Then results (i.., de-

flections and strsses) for several laminated shell problems are presented.

The results for laminated shells should serve as references for future inves-

tigations.

All of the results reported here were obtained using the double-precision

arithmatic on an IBM 3081 processor. Most of the sample problems were an-

alyzed using a 2 x 2 uniform mesh of the nine-node (quadratic) isoparametric

rectangular element.

1. Bendin 9 of a simply supported plate strip (or, equivalently, a beam) under

uniformly distributed load.

The problem is mathematically one-dimenSional and an analytical solution

of the problem, based on the classical theory, can be found in Timoshenko and

Womowsky-Krieger [28]. The plate length along the y-coordinate is assumed to

be large compared to the width, and it is simply supported on edges parallel

12

to the y-axis,

assumed:

The following simply supported boundary cpnditions are

w = _2 = 0 along edges x = -+ 127mm (21)

All inplane displacement degrees of freedom are restrained, A 5 x I mesh of

four-node rectangular elements in the half plate is used to analyze the prob-

lem. The data and results are presented in Fig. 2. The present result is in

good agreement with the analytical solution.

2, Clamped square plate under uniform load.

Due to the biaxial symmetry, only one quadrant of the plate is modelled

with the 2 x 2 mesh of nine-node elements (4 x 4 mesh of linear elements give

almost the same result). Pertinent data and results are presented in Fig. 3

for side to t_ickness ratios a/n-= 10 and 500. The result for a/h = 500 is in

agreement with the resul:s of Way [29], The difference is attributed to the

fact that the present model includes the inplane displacement degrees of free-

dom and transverse shear deformation,

Figure 4 contains transverse deflection versus load for clamped ortno-

tropic, cross-ply, and angle-ply plates. The lamina properties are

EI = 25 x 10_ N/ram2, E2 : 2 x i0 L_ N/ram2, GI2 = G13 = 10L_ N/ram2

G23 = 0.4 x 10" N/mm 2, v12 : 0.25.

For the same total thickness the clamped orthotropic square plate is stiffer

than both two-layer angle-ply and cross-ply plates.

13

Load

qo (N/ram2)

20

16

12

4

_-_,_ ' ' ' ' ' ' '/_

l lT oore,e., /,'I i _1 __.analytical[28} t/

./

0 5 10 15 20 25

Deflection, -w (in ram)

Figure 2. Bending of an isotropic simply supported

plate strip under uniform load.

14

Load

qo a_

120

i00

80

60

40

20

0

0.0

J alh : 10

._. alh = 500

.... Way [29]

E : 0.2 x 10 s NImm2

:0.3

a : I000 mm

_.. Linear (_ =f

a I0)Linear (_ =

0.4 0.6 0.8

Deflection,-w/h (-w/h)

Figure 3. Bending of clamped isotropic square plate underuniform load.

15

Load

qo

( lO-4N/mm 2 )

20

16

12

0

0.0

1 i L |

-o- orthotropic

--o-- [-45°/45 ° ]

.-,_...-. {0°/90 o]

I I

d

sAS E1 : 25x10_N/mm 2f AS"

E2 : 2 xlO_N/mm 2

GI2 = GI3 : I04N

G23 : O.4xlO4NI I I

1.0 2.0 3.0 4.0

Deflection, -w (in mm)

5.0 6.0

Figure 4. Bending of clamped orthotropic and laminated

square plates under uniform load.

16

3. S!mply supported, isotropic spherical snell under point load.

The pertinent data of the shell is shown in Fig. 5. A uniform mesh of 2

by 2 quadratic elements is used in a quadrant. The effect of three types of

simply supported conditions on the center deflection and center normal stress

is investigated:

SS-I: u = w = ¢i: 0 at y : b; v : w = ¢2 = 0 at x = a

SS-2: u = v = w = @i= 0 at y = b; u = v = w = Cz= 0 at x = a(22)

SS-3: v = w = ¢_= 0 at y = b; u = w = ¢2 = 0 at x = a

Table I contains the results for the three boundary conditions. It is clear

from the results that all three boundary conditions give virtually the same

results for a/h = 160, and differ significantly (especially SS-I differs from

both SS-2 and SS-3) for a/h = 16. Thus, the effect is more in thick shells

than in thin shells. The stress _x shown in Fig. 5 is evaluated at point x =

y = 1.691" in the top layer

4. Simply supported isotropic cylindrical shell under point load.

The geometry and finite-element mesh of the shell are shown in Fig. 6.

Once again, the effect of various simply supported boundary conditions (22) on

the deflections and stresses for the problem is investigated using a uniform

mesh of 2 x 2 quadratic elements. The results are presented in Table 2. For

the geometry and loading used here (R = 2540, a = 254, h = 12.7), the boundary

conditions have very significant effect on the solution. Boundary conditions

SS-2 and SS-3 give al_nost the same results whereas SS-I gives about 2-i/2

times the deflection given by SS-2 or SS-3 boundary conditions.

17

Table I. Effect of various simply supported boundary conditions on the centerdeflections and normal stress in spherical shells under point load(E = i0 ? psi, v = 0.3).

Load Solution SS-I SS-2 SS-3P/n 2 aln=160 a/h:16 alh:160 a/h=16 alh:160 alh:16

4,000 -w* 0.0155 - 0.0152 - 0.0152 -

"Ox* 893 - 984 - 894 -8,000 -w 0.0329 0.0349 0.0324 0.0255 0.0324 0.0258

"°x 1,880 6,535 1,882 6,015 1,882 6,03112,000 -w 0.0529 - 0.0522 - 0.0521 -

-o 2,980 - 2,985 - 2,986 -

16,000 -wx 0.0760 0.0793 0.0752 0.0520 0.0751 0.0525

"°x 4,220 13,230 4.228 12,200 4,229 12,24020,000 -w 0.1038 - 0.1028 - 0.1027 -

-o 5,657 - 5,671 - 5,672 -

2¢,000 -wx 0.1364 0.1083 0.1354 0.0792 0.1353 0.0800

"_x 7,268 20,110 7,289 18,500 7,291 18,55028,000 -w 0.1761 - 0.1752 - 0.1751 -

"_x 9,128 - 9,160 - 9,162 -32,000 -w 0.2234 0.1472 0.2227 0.1072 0.2227 0.1083

-oX 11,180 27,170 11,220 24,930 11,230 25,000

* w(O,O), _x(A,A); A = 1.691

Table 2. Effect of various types of simply supported boundary conditions onthe deflections and stresses of anisotropic cylindrical shell under

point load.

Load ,P(N)

SS-I SS-2 SS-3

-w(mm) -_y(N/mm 2) -w -Oy -w -Oy

25O5O0

750

1,000

I,250

1,500

1,750

2,000

2.5804(2) 2.868 0.6544(4) 1.706 0.6698(4) I5.1626(2) 5.713 1.3533(4) 3.478 1.3843(4) 37.7343(2) 8.506 2.1057(4) 5.327 2.1522(4) 5

10.278(2) 11.210 2.9234(4) 7.265 2.9855(4) 712.733(2) 13.80 3.8241(4) 9.312 3.9017(4) 915.204(2) 16.25 4.8349(4) 11.50 4.9279(4) Ii17.560(2) 18.560 6.0331(5) 13.91 6.1423(5) 1319.843(2) 20.730 7.5316(6) _ 16.66 7.6610(6) 16

.706

.477

.321

.242

.288

.46

.85

.57

18

O. 30

0.25

0.20

Deflection

w(in.)

0.15

0. i0

0.05

I I I l* "I ' I ' • ,it

_ ,,I. \ /,i

sit/_ -.0- -W

A=(].69l,I.691)

0 80 160 240 320

Load, P (Ibs)

12,000

i0,000

9,000

o (psi)x

6,000

4,000

2,000

Figure 5. Bending of a simply supported (SS-3),

isotropic, spherical shell under pointload.

19

e e Material 2/

R

Y

Figure 6. Geometry of a cylindrical shell.

2O

5. Clamped isotropi c cxlindrical shell under uniform loa_ing.

Figure 7 contains the pertinent data and results for a clamped cylindri-

cal snell {isotropic) subjected to uniform load. The results are compared

with those obtained by Dnatt [30]. The agreement is very good,

6. Clamped ortnotropic cylindr!cal shell subjected to internal pressure.

Figure 8 contains the geometry and plots of center deflection and center

stress versus the internal pressure for the promlem. The orthotropic material

properties used in the present study are:

E1 : 7.5 x 106 psi, E2 = 2 x 106 psi, GI2 : G13 : G23 : 1.25 x 106 psi

vl2 : o.25 (23)

The present result, obtained using the 2 x 2 mesh of quadratic elements, is in

excellent agreement with that obtained by Chang and SawamipnaKdi [31].

7. Nine-layer [0°/90°/0°.../0°_ cross-ply spherical shell subjected to

uniformIx distributed load.

The following geometrical data is used in the analysis (with SS-3 boundary

conditions):

R1 = R2 = R = 1,000 in., a = b = 100 in., h = i in. (24)

Individual layers are assumed to be of equal thickness (hi = h/9), with the

zero-degree layers being the inner and outer layers. The following two sets

of orthotropic-material constants, typical high modulus graphite epoxy materi-

al (the ratios are more pertinent here), for individual layers are used:

21

300O

2500

2000

qo(lO'6N/mm2)

1500

1000

500

-W

--o-- -Oy (26.84,26.84)

___ Dhatt [30]

!

/I

!

0 2

R : 2540 mm

a : 254 mm

e : 0.1 rad

v - E :_3103 N/ram2

v: 0.3

4 6 8 10 12

2 3 4 5 6

Deflection and Stress

-w(mm)

-o (N/mm 2)Y

Figure 7. Bending of a clamped, isotropic, cylindrical shell underuniform load.

22

20.0

16.0

12.0

Load

qo(ksi )

8.0

o present 1_ Reference [311

W

o Ox at x = y = 1.057

shell thickness = l"

0.00 4 8 !2 16 20 24

0 50 100 150 200 250 300

w(in.)

_(ksi)

Figure 8. Bending of a clamped orthotropic cylindrical shell

subjected to internal pressure.

23

Mat.-l: E1 : 25 x 106 psi, E2 : 106 psi, G12 : GI3 : 0.5 x 106 psi

G23 : 0.2 x 106 psi, v12 = 0.25 (25)

Mat.-2: EI = 40 x 106 psi, E2 = 106 psi, GI2 = GI3 : 0.6 x 106 psi

G23 = 0.5 x 106 psi, v12 = 0.25 (26)

Figure 9 contains plots of center deflecton (w/h) versus the load parameter

IP = qoR2/E2 h2) for the two materials. Snell constructed of Material 1

deflects more, for a given load, than the shell laminated of Material 2

(because Material 2 is stiffer), and consequently experiences greater degree

of nonlinearity. Note that the difference between the nonlinear deflections

of the two smells increase nonlinearly, indicating that the shell made of

Material 2 can take much more (ultimate) load than apparent from the ratio of

_2)IE(1)moduli of the two materials, E i "

8. Effect of various simpIx-supported boundarx conditions on the deflections

of two-layer cross-pIx spherical shells under uniform load.

As pointed out in Problems 3 and 4, the transverse deflection is sensi-

tive to the boundary conditions on the inplane displacements of simply sup-

ported shells. To further illustrate this effect for laminated shells, a set

of four types of boundary conditions are usea, and the results are presented

in Table 3. Here SS-4 has the following meaning:

w = ¢i = 0 on x = aSS-4 (27)

w = ¢ : 0 on y : b2

24

Load

- qoR2P --

7

6

4

0

;rial 2

I

0.0

_Material 1

R : 1000 in.

f / a = b : 50 in.

| I I I 1 I i

0.2 0.4 0.6 0.8

Deflection, (-w/h)

Figure 9. Bending of nine-layer cross-ply

[0o/90o/0o/...1 spherical shell

subjected to uniformly distributed

load.

25

Table 3. Effect of various simply-supported boundary conditions on the trans-verse deflections of cross-ply [0o/90 °] spherical shells under

uniform load (Material l; shell dimensions are the same as those in Fig.

qo -w (in.)SS-I SS-2 SS-3 SS-4

(psi)

0.50 0.3344 0.04246 0.04257 0.45920.75 0.5757 0.06599 0.06617 0.8255

1.00 0.9485 0.09144 0.09171 1.3845

1.25 1.6529 0.11926 0.11966 1.95891.50 2.2826 0.15008 0.15063 2.3597

1.75 2.6421 0.18478 0.18556 2.5951

2.00 2.8499 0.22473 0.22584 2.8074

2.25 3.0764 0.27425 0.27593 3.0284

2.50 3.2432 0.33534 0.33795 3.1948

2.75 3.4214 0.42970 0.43487 3.3719

26

Onceagain we note that SS-2 and SS-3 give almost the sa_e deflections.

Boundary conditions SS-I and SS-4 give deflections an order of magnitude high-

er than those given by SS-2 and SS-3. Thus, boundary conditions SS-2 and SS-3

make the shell quite stiffer.

9. Two-layer cross-ply [0°/90 °] and angle-ply [-45°/45°] r simply-supported

(SS-3) spherical shells.

Figure 10 contains the pertinent data and results (with different scales)

for the cross-ply and angle-ply shells (of Material 2). It is interesting to

note that the type of nonlinearity exhibited by the two shells is quite dif-

ferent; the cross-ply shell gets softer whereas the angle-ply shell gets

stiffer with an increase in the applied load. While both shells have bending-

stretching coupling due to the lamination scheme (B22 = - Bli nonzero for the

cross-ply shell and B16 and B26 are nonzero for the angle-ply shell), tne

angle-ply experiences shear coupling that stiffens the spherical shell rela-

tively more than the normal coupling (note that, in general, shells get softer

under externally applied inward load).

Figure 11 contains plots of center deflection, normal stress I-Oyl and

shear stress _yz) at x = y = 5.283" versus load for two-layer cross-ply

(0°/90 °) spherical shell (Material I) under point load at the center of the

shell. The nonlinearity exhibited by the stresses (especially _ ) is lessyz

compared to that exhibited by the transverse deflection.

10. Two-layer clamped cylindrical shells under uniform loads.

Figures 12 and 13 contain results (i.e., w, _y, _xz versus load) for

cross-ply [0°/90 °] and angle-ply [-45°/45 °] clamped cylindrical shells under

uniform load. The load-deflection curve for the cros-ply snell resembles that

27

cross-ply

Load,

qo(pSi)

0.0

3"01

2.5

2.0

1.5

l.O

0.5

0.0

0.0

Deflection, -w (lO'2in)

O.l 0.2 0.3

I I I I I

a=b= 50in

h= lin.

O.l 0.2 0.3

Deflection, w (in in.)

angl e-ply

0.4 O.S

, , angle-nly

l

Material 2

0.4

/90o3 0.5

0.4

Load, no(pSi)

0.3

0.2

O.l

0.0

0,5 --cross-ply

Figure lO. Bending of two-layer cross-ply and angle-ply,simply supported (SS-3) spherical shells underuniform load.

28

P(lO31bs)

I0

8

2

o -w(in.)

a -Oy(5.283,5.283)(10"psi)

-_ (5.283,5.283)(I02psi)yz

0 | !

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Deflection and Stresses, -w,-ay,-_y z

Figure II. Bending of a cross-ply {0°/90 °] spherical shell

(SS-3, Material I), under point Ioad.(see Fig. lO

for the shell dimensions)

29

Load

qo (102 )

0.8

0.6

0.4

0.2

0.0

! I | I | I I ! I ! I

IO-W _-

o-oy(lO3pSi)atA-_ at B _o _ ,,o- .......tY'"

yz _ .a'"- ,,_..'"" -

A:(26._4,26.84_/.o.--._..-_ B=(100.2, S o'"-m'""--

100.2y , ,_?:._." "'-- _ -

,2 ,-°....._'"# o" ..._-

/._'c_""_"R = 2540 in. ___'a-' e = 0.1:'."'" a-254 in. Y" l':"a-' _ := 2.54

X"

• | I ,_ l I I I 1 , I _ 1 ,

0.0 0.4 0.8 1.2 1.4 1.6 -(w)

0.0 0.5 1.0 1.5 2.0 2.5 -(OyX103)

0 5 10 15 20_(Cyz)

Figure 12. Bending of a clamped angle,-ply [-45"/.45"] cylindricalshell under unifprm load. {Material l)

30

Load

no (psi)

2.0

1.6

1.2

0.8

0.4

0.0

0.0 1.0 2.0 3.0 4.0 5.0

0.0 0.4 0.8 1.2 1.6 2.0

0 20 40 60 80 i00

-W

-oY

XZ

Figure 13. Bending of a clamped cross-ply [0°/90 °) cylindricalshell under uniform load (Material l)

31

of the isotropic shell in Fig. 7, but exhibits greater degree of nonlinearity

(being stiffer). The angle-ply shell exhibits different type of nonlinearity

(softening type) for all loads.

Ii. quasi-isotropic, clamped r cylindrical shell under uniform load.

Two types of quasi-isotropic clamped cylindrical shells are analyzed:

o 4 o o oType I: [0 / 5 /90 /-45 ]sym.

Type 2: [O°/+45°/90]sym.

(28)

Material I properties are assumed for each lamina (8 layers). The geometric

data and results are presented in Fig. 14. Compared to the results presented

in Figs. 12 and 13, the quasi-isotropic shells nave the 'near-inflection'

point at higher loads; the load-deflection curve has essentially the same form

as Chat of the cross-ply shell (see Fig. 12).

CONCLUSIONS

A shear-flexible finite element based on the shear deformation version of

the Sanders' theory and the von Karman strains is developed, and its applica-

tion to isotropic, ortnotropic, and laminated (cross-ply and angle-ply) snells

is illustrated via numerous sample problems. Many of the results, especially

those of laminated shells, are not available in the literature and therefore?

should serve as references for future investigations. From the numerical com-

putations it is observed that boundary conditions on the inplane displacements

have significant effect on the shell def|ections and stresses. Also, it is

noted that the for;n of nonlinearities exhibited by different lamination schemes.

32

2.0

1.6

Load 1.2

qo (psi)

0.8

0.4

}

0 -W 1a _oy(lO 4) at A Type 1

Oxz(i02) atBJ

.... w Type 2

A=(26.84,26.84), B=(227.2,26.84)

qo

R = 2540 in.

a = 254 in.

h = 2.54 in.

0.0

0.0 1.0 2'0. 0 .0 -w

0.0 0.5 1.0 1.5 2.0 2.5 -_y

0.0 0.2 0.4 0.6 0.8 1.0 1.2 Oxz

Figure 14. Bending of clamped quasi-isotropic cylindrical shellsunder uniform load.

33

I •

•

•

•

•

•

•

•

•

I0.

11.

12.

13.

14.

REFERENCES

P. M. Naghdi, "A Survey of Recent Progress in the theory of E]astic

Shells," Appl. Mech. Reviews, Vol. 9, No. 9, pp. 365-368, 1956.

C. W. Bert, "Analysis of Sbells,"Analxsis and Performance of Composites,L. J. Broutman (ed.), Wiley, New York, pp. 207-258, 1980.

A. E. H. Love, "On the Smal| Free Vibrations and Deformations of the

Elastic Shells," Phil. Trans. Rox_ Soc. (London), Ser. A, Vol. 17, pp.

491-546, 1888.

E. Reissner, "A New Derivation of the Equations for the Deformation of

Elastic Shells," Am. Jour. Math., Vol. 63, No. i, pp. 177-184, 1941.

E. Reissner, "On Some Problems in Shell Theory," Structural Mechanics

(Proceedings of the First Symposium on Naval Structural Mechanics), J. N.

Goodier and N. J. Hoff (eds.), Pergamon Press, New York, pp. 74-114,1960.

E. Reissner, "A Note on Generating Generalized Two-Dimensional Plate and

Shell Theories," J. Appl. Math. Phxsics (ZAMP), Vol. 28, pp. 633-642,1977.

J. L. Sanders, "An Improved First-Approximation Theory for Thin Shells,"NASA Tecnnical Report R-24, 1959.

S. A. Ambartsumyan, "Calculation of Laminated Anisotropic Shells,"

Izvestiia Akademiia Nauk ArmensKoi SSR_ Ser. Fiz. Mat. Est. Tekh. Nauk.,Vol. 6, No. 3, p. 15, 1953.

S. A. Ambartsumyan, Theorx of Anisotropic Shells, Moscow, 1961; Englishtranslation, NASA TT F-118, May 1964.

S. B. Dong, K. S. Pister, and R. L. Taylor, "On the Theory of Laminated

Anisotropic Shells and Plates," Journal of Aerospace Sicneces, Vol. 29,p. 969-975, 1962.

L. H. Donnell, "Stability of Thin Walled Tubes in Torsion," NACA Report479, 1933.

G. E. O. Widera, and S. W. Chung, "A Theory for Non-Homogeneous

Anisotropic Cylindrical Shells," Journal of Applied Mechanics (ZAMP),Vol. 21, pp. 378-399, 1970.

J. A. Zukas, and J. R. Vinson, "Laminated Transversely Is•tropic

Cylindrical Shells," journal of Applied Mechanics, pp. 400-407, 1971.

S. B. Dong, and F. K. W. Tso, "On a Laminated Orthotropic Shell Theory

Including Transverse Shear Deformation," Journal of Applied Mechanics,Vol. 39, pp. 1091-1097, 1972.

34

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

J. M. Whitney, and C. T. Sun, "A Refined Theory for _amianted Anisotrop-ic, Cylindrical Shells," Journal of Applied Mechanics, Vol. 41, pp. 471-476, 1974.

G. E. O. Widera, and D. L. Logan, "Refined Theories for Nonhomogeneous

Anisotropic Cylindrical Shells: Part I-Derivation," Journal of the Engi-

neerin_ Mechanics Division, Vol. 106, No. EM6, pp. 1053-1074, 1980.

D. L. Logan, and G. E. O. Widera, "Refined Theories for Nonhomogeneous

Anisotropic Cylindrical Shells: Part II-Application," journal of the En-

_ineerin9 Mechanics Division, Vol. 106, No. EM6, pp. 1075-1090, 1980.

, . "Finite Element Analysis of SandwichL. A. Schmit and G R. Monforton,

Plate and Laminate Shells with Laminated Faces," _nerican Institute of

Aeronautics and Astronautics Journal, Vol. 8, pp.1454-1461, 1970.

S. C. Panda, and R. Natarajan, "Finite Element Analysis of Laminated

Shells of Revolution," Computers and Structures, Vol. 6, pp. 61-64, 1976.

K. N. Snivakumar, and A. V. Krishna Murty, "a High Precision Ring Element

for Vibrations of Laminated Shells," journal of Sound and Vibration, Vol.

58, No. 3, pp. 311-318, 1978.

K. P. Rao, "A Rectangular Laminated Anisotropic Shallow Thin Shell Finite

Ele_nent," Computer Methods in Applied Mechanics and En_ineerin_,Vol. 15,1978, pp. 13-33.

• . "Finite Element Analysis of LaminatedP Siede, and P. H. H Chang,

Plates and Shells," NASA CR-157106, 1978 (132 pages).

Y. S. Hsu, J. N. Reddy, and C. W. Bert, "Thermoelasticity of Circular

Cylindrical Shells Laminated of Bimodulus Composite Materials," Journal

of Thermal Stresses, Vol. 4, No. 2, pp. 115-177, 1981.

J. N. Reddy, "Bending of Laminated Anisotropic Shells by a Shear Deform-

able Finite Element," Fibre Science and Technologx, Vol. 17, pp. 9-24,1982.

A. VenKatesh and K. P. Rao, 'A Doubly Curved Quadrilateral Finite Element

for the Analysis of Laminated Anisotropic Thin Shells of Revolution,"

Computers and Structures, Vol. 12, pp. 825-832, 1980.

A. Venkatesh and K. P. Rao, "Analysis of Laminated Shells with Laminated

Stiffners Using Rectangular Shell Finite Elements," Computer Methods inAppl. Mech. Engng., Vol. 38, pp. 255-272, 1983.

J. N. Reddy, "Exact Solutions of Moderately Thick Laminated Shells," j.

Engineerin 9 Mechanicsr. ASCE, to appear.

S. Timoshenko and S. Woinowsky-Krieger, Theor_ of Plates and Shells, 2rided., McGraw-Hill, New York, 1959.

S. Way, "Uniformly Loaded, Clamped, Rectangular Plates with Large Defor-

mation," Proc. 5tn Inter. Contr. of Appl. Mech., Cambridge, MA, 1938.

35

30. G. S. Dhatt, "Instability of Thin Shells by the Finite Element Method,"IASS Sxmposium for Folded Plates and Prismatic Structures, Vienna, 1970.

31. T. Y. Chang and K. Sawamiphakdi, "Large Deformation Analysis of Laminated

Snels by Finite Element Method," Computers and Structures, Vol. 13, pp.

331-340, 1981.

36

APPENDIX I "

Stiffness Coefficients:

I _u3 1 _u3

Let fl - 2 _xI ' f2 : 2 ax2

[K11] : A11[SZl ] + A16([S12 ] + [$21]) + A66[S22]

- CoCBI6([SI2 ] + [S2Z]) + 2B66[$22] - CoD66[S22] ) + A5_5 [SOO]

[KI2] : A12[S12 ] + A16[$II ] + A26[$22 ] + A66ES21 ]

A45

- COCB26[$22 ] - B16[$11 ] ÷ COD66[$21] ) + R-_" 2 IS 00]

[KI3] : flCAII[S11] + AI6[S12] + [$21]) + A66[S22] )

+ f2(AI2[S12] + AI6[SII] + A26[$22 ] + A66[$21] )

I + i (AI2[SIO ] + A26[$2o] )+ _ (All[S_°]+ AI6[s2O])

B26 s2O] I S02] A55[SO I+R]- [ ) " (A4 E + ])

- Co[fI(B16[S21 ] + B66[$22] ) + f2(B26[S22] + B66[$21])]

[K I"] : B11[SII ] +B16([S12] • IS21]) + B66[S22]

" c°(D16[$21] + D66[$22]) " '_-I A55[S°°]

37

[KI5] = B12[S12] + B16[SII] + B26[S22] + B66[S21] ..

" c°CD25[$22] + D66[S21]) " _I A45[S°°]

[K21] : [KI2] T

[K22] : A22[$22] + A26([SI2] + [$21]) + A66[SII] + 2COB66[SII]

+ Co{B26([S12] + [$21]) + COD66[S11] ) - _A_4 [sOO]

[K23] : flIAI2[S21] + A26[$22] + AI6[SII] + A66[S12] )

+ f2(A22[S22] + A26([S21] + [$12]) + A66[SII] )

+ R_CAI2[s_°]÷ A!6FS_°])+R_CAn[S2°]" A26[S_°])

rB16 B26_[s_O] I sO2] A45[sO_+ Co ',RI + R2 J - R"2"{A44[ + ])

+ Co[fI(B16[SII] + B66[SI2] ) + f2(B26[SI2] + B66[SII])]

[K2"] : BI2[S21] + B26[S22] + BI6[SZl] + B66[SI2]

+ c°(DI6[SII] + D66[SI2]) -_2 A45[S'0°]

[K25] : B22[$22 ] + B26([S21] + [S12]) + B66[SII]

+ CoCD26[SZ2] + D66[$11]) - _2 A44[S 0°]

38

TCK31]N L : [2KI3]N L , NL : Nonlinear portion of the matri,x

[K32]N L = [2K23]_ L

[K33] = A45[S12 ] + A55[$II] + A44[$22] + A45[S21]

÷ 2[s_](A11f_ _.zA16flf2 ÷ A66f_)

+ 2C[S 12] + [S21])[f_A16 + (A12 + A66)flf2 + f_A26 ]

+ ZEs22]CA66f_÷ ZA26flf2 • A22f_)

L RI R2J R_2 ' RI R2 J;"i

.,AllA12 ÷f2 A16+A26 ]+ CES01] + 2[SI°])[_ILTI + R2J ' RI R2J

+ (IS 02] + 2[$2°])[ i_ RI R2J _ RI+ + _'j j

[K 3W] = A55[SI° ] + A45[S20 ]

+ 2f1(B11[S11] + B16([$12] + [S21]) + B66[S22] )

+ 2f2(BI2[S21] + B66[SI2] + B26[S22] + B16[$II] )

39

[K 35] : A45[sIO ] + A44[S2°]

+ 2fl(B_2[SZ2] + B66[S21] + BI6[SlI] + B26[S22] )

+ 2f2(B22[S22] + B26 [S 12] + B26[S21] + B66[S11] )

RI R2J _ RI "2

Z [K3.]_[KWI] : [KI4]T' [K42] : [K24]T' [K43]NL : 2 L

[K _"] : Oll[Sll] + O16([$12] + [$21]) + D66[S22] + A55[S°°]

[K"5] : O12[$12 ] + 016[$11] + D26[$22] + D66[$21] + A45[S0°]

CK51] : [KIS]T' [K52] : [K25]T' [KS3]NL : 2-[K3S] L

"r

[K 5_] = [K"5]'

[K sS] : D22[$22 ] + D26([S12] + [$21]) + O66[S11] + A44[sO0]

[K:B]Linear = [KB:]_inear

where

=B Odxi O@j sO.°. _jdXldX2sij : I _ dxI_x2 : I _iOx@ . ' Ij eQe Ox

4O

It should be noted that although fl and f2 are shownfactored outside the

matrices, in the evaluation of the coefficients by the Gaussquadrature fl and

f2 are considered as parts of the integrals. For exan_)le fIA11[S 11] is

evaluated byN N Bu3

• Z A11[(_T1)*i_j]Xl=Z ,x2:zjWIWjdetJoS fiA11_i_j_x1dx_:½ I!IJ:l IQe

where N is the number of Gauss points, WI and Wj are the Gauss weights, ZI and

Zj are the Gauss points, and Jo is the jacobian of the transfomation.

WP:jNRKCI

41

PART 2

ANALYSIS OF LAMINATED COMPOSITE SHELLS

USING A DEGENERATED 3-D ELEMENT

W. C. Chao* and J. N. Reddy

Department of Engineering Science and Mechanics

(Thi_ pap_ _5 to appear in Int. Jow_nal o_ Numerical Methods in Engng. )

SUMMARY

A special three-dimensional element based on the total Lagrangian

description of the motion of a layered anisotropic composite medium is

developed, validated, and employed to analyze laminated anisotropic

composite shells. The element contains the following features:

geometric nonlinearity, dynamic (transient) behavior, and arbitrary

lamination scheme and lamina properties. Numerical results of nonlinear

bending, natural vibration, and transient response are presented to

illustrate the capabilities of the element.

INTRODUCTION

Composite materials and reinforced plastics are increasingly used

in automobiles, space vehicles, and pressure vessels. With the increased

use of fiber-reinforced composites as structural elements, studies

involving the thermomechanical behavior of shell components made of

composites are receiving considerable attention. Functional

requirements and economic considerations of design have forced designers

to use accurate but economical methods of determining stresses, natural?

frequencies, buckling loads etc.

Graduate Research Assistant; presently at the University of DaytonResearch Institute

42

Majority of the research papers in the opeh literature on shells

is concerned with bending, vibration, and buckling of isotropic

shells. As composites materials are making their way into many

engineering structures, analyses of shells madeof such materials

becomesimportant. The application of advanced fiber composites in jet

engine fan or compressor blades and high performance aircraft require

studies involving transient response of composite shell structures to

assess the capability of these materials under dynamic loads.

Finite-element analysis of shell structures in the past have used

one of the three types of elements: I. a 2-D element based on a two-

dimensional shell theory; 2. a 3-D element based on three-dimensional

elasticity theory of shells; and 3. a 3-D degenerated element derived

from the 3-D elasticity theory of shells. The 2-D shell theory is

derived form the three dimensional continuum field equations via

simplifying assumptions. The simplifications require the introduction

of the static and kinematic resultants, which are used to describe the

equations of motion. The unavailability of a convenient general

nonlinear 2-D shell theory makes the 2-D shell element restrictive in

its use. The degree of geometric nonlinearity included in the 2-D shell

element is that of the yon Karman plate theory. In contrast to the 2-D

shell theory, no specific shell theory is employed in the 3-D

degenerated element; instead, the geometry and the displacement fields

are directly discretized and interpolated as in the analysis of

continuum problems.

Finite-element analyses of the large-displacement theory of solids

are based on the principle of virtual work or the associated principle

of stationary potential energy. Horrigmoe and Bergan [l] presented

43

classical variational principles for nonlienar problems by considering

incremental deformations of a continum. A survey of various principles

in incremental form is presented by Wunderlich [2]. Stricklin et al.

[3] presented a survey of various formulations and solution procedures

for nonlinear static and dynamic structural analysis. The formulations

include the pseudo force method, the total Lagrangian method, the

updated Lagrangian method, and the convected coordinate method.

The only large-deflection analyses of laminated composite shells

that can be found in the literature are the static analysis of Noor and

Hartley [4] and Chang and Sawamiphakdi [5]. Noor and Hartley employed

the shallow shell theory with transverse shear strains and geometric

nonlinearities to develop triangular and quadrilateral finite

elements. Chang and Sawamiphakdi presented a formulation of the 3-D

degenerated element for geometrically nonlinear bending analysis of

laminated composite shells. The formulation is based on the updated

Lagrangian description and it does not include any numerical results for

laminated shells.

From the review of the literature it is clear that first, there

does not exist any finite-element analysis of geometrically nonlinear

transient response of laminated anisotropic shells, and second, the 3-D

degenerated element is not exploited for geometrically nonlinear

analysis of laminated anisotropic shells. In view of these

observations, the present study was undertaken to develop a finite-

element analysis capability for the static and dynamic analysis of

geometrically nonlinear theory of laminated anisotropic shells. A 3-D

degenerated element with total Lagrangian description is developed and

used to analyze various shell problems.

44

INCREMENTAL_ TOTAL-LAGRANGIAN FORMULATION OF A CONTINUOUS MEDIUM

The primary objective of this section is to review the formulation

of equations governing geometrically nonlinear motion of a continuous

medium. In the interest of brevity only necessary equations are

presented. For additional details the reader is referred to References

[6-lO].

We describe the motion of a continuous body in a cartesian

coordinate system. The simultaneous position of all material points

(i.e., the configuration) of the body at tlme t is denoted by Ct,

and CO and Ct+at denote the configurations at reference time t = 0 and

time t + at, respectively (see Fig. l). In the updated Lagrangian

description all kinetic and kinematic variables are referred to the

current configuration at each time and load step. In the total

Lagrangian description all dependent variables are referred to the

reference configuration. The updated Lagrangian is more suitable for

motions that involve very large distortions of the body (e.g., high-

velocity impact). The total Lagrangian is more convenient for motions

that involve only moderately large deformations. In the present study

the total Lagrangian formulation is adopted.

Here we present a derivation of the equilibrium equations at

different time steps using the total Lagrangian approach. The

coordinates of a typical point in Ct is denoted by tx , (txl,tx2 tx3).

The displacement of a particle at time t is given by

tu = tx - °x or tu =- - - i tx i -°x i (1)

The increment of displacement during time t to t + Lt is defined by

ui t+atu I (2)= . - tu i

45

xI

x3

/

C0

Ct+6t

/

Vt, At )

V = Volume

A = Area

x2

Figure 1 Motion of a continuous body in Cartesian coordinates

46

The principle of virtual displacements can be:employed to write the

equilibrium equations at any fixed time t. The principle, applied to

the large-displacements case, can be expressed mathematically as

t+Atu i +[ % 6ui dVo [Vo Vo

t+At$ (t+atij 6 _ij)dVo

= _ t+atT i 6uidA o +A V0 0

oo t+atF i 6ui dVo (3)

where summation on repeated indices is implied; Vo, Ao, and 00 denote,

respectively, a volume element, area element, and density in the initial

configuration, $ij are the components of second Piola-Kirchhoff stress

tensor, ¢ij the components of Green-Lagrangian str_ntensor, T i the

components of boundary stresses, and Fi are the components of the body

force vector; the superposed dots on ui denotes differentiation with

respect to time, and 6 denotes the variational symbol. In writing Eq.

(3) it is assumed that ¢ij is related to the displacement components by

the kinematic relations

= ½ (t+Atu t+Atu. +t+Atcij i,j + 3,it+Atu t+At u )

m,i m,j (4)

where ui, j BUi/BX j The strain components t+At= . Eij can be expressed in

terms oF current strain and incremental strain components as

I (tuit+At¢i j = _ ,j + tu tu . tu j)J,i + m,l m,

+ 1 (ui,j + uj, it U

m,i Um,j + Um,i turn,j) + Um,i Um,j+

t- ¢ij + (eij + hij) (5)

47

where eij and nij denote the linear and nonlinear incremental strains.

The stress components t+atsij can be decomposed into two parts:

t+atsij = tsij + Sij (6)

where Sij is the incremental stress tensor. The incremental stress

components Sij are related to the incremental Green-Lagrange strain

components, ¢ij = eij + niJ' by the generalized Hooke's law:

Sij = CijkzCk_,

where Cijk_ are the components of the elasticity tensor.

(7), Eq. (3) can be expressed in the alternate form

(7)

Using Eq. (4)-

t+At ""oo Ul6U dVo +

V i V0 o

Cijk_(ek_6nij + nk_6eij)dV o

+ f tsij _eij dVo = 6W -

Vo Votsij 6nij dVo (8)

where _W is the virtual work due to external loads.

FINITE-ELEMENT MODEL

Geometry of the Element

Consider the solid three-dimensional element shown in Fig. 2.

The coordinates of a typical point in the element can be written as

n n 12{ xj: + z _j(_l,_2)-xi z _j(_i,_2) _ (x_)top J:lj=l ( i)b°ttom

: (g)

where n is the number of nodes, Vi(Cl,C2) are the finite-element

interpolation (or shape) functions, which take in the element, the value

of unity at node i and zero at all other nodes, _I and _2 are the

normalized curvilinear coordinates in the middle plane of the shell,

and _ is a linear coordinate in the thickness direction and x , x , and x3

48

Figure 2 Geometry of the degenerated three-dimensional element

49

are the global coordinates at node i. Here {i,{2, and _ are assumed

to vary between -l and +l. Now let (see Fig. 2)

i i i

V3k = (Xk)to p - (Xk)bottom (lO)

where V_k is the k-th component of the vector v_. Then Eq. (9) becomes

n A •

xi = z [_j(X_)mid + _j _ hj e_i]j=l

(ll)

where hj is the thickness of the element at node j. For small

deformation, the displacement of every point in the element can be

written as

nz J + h ej ej ej oJ_] (12)ui: 2( Ii2- 21 I,j=l

where 0 and e2 are the rotations about (local) unit vectors eI and ,

respectively, ul, u2, and u3 are the displacement components

corresponding to the global coordinates xl, x2, x3 directions

respectively, and uil, u_ and uI are the values of the displacements

(referred to x) at node i. In writing Eq. (12), we assumed that a line

that is straight and normal to the middle surface before deformation is

still straight but not necessarily 'normal' to the middle surface after

deformation. The strain energy corresponding to stress perpendicular to

the middle surface is ignored to improve numerical conditioning when the

three dimensional element is employed. This constraint corresponds only

50

to a part of the usual assumptions of a two-dimensional shell theory.

The relaxation of the requirement that straight lines perpendicular to

the middle surface remain normal to the deformed middle surface permits

the shell to experience shear deformation - an important feature in

thick shell situations.

Displacement Field in the Element

present study the current coordinates txi are interpolatedIn the

by the expression

txl j-I

and the displacement by

= _ _j [tu_ + ½ {hj (te_l- °e_l)] (14)tui j=l

n .t+At_j

ui = j=IZ _j[u_ + ½ chj ( e3i - te_l)) (15)

Here tu_ and u_ denote, respectively, the displacement and incremental

displacement components in the xi-direction at the J-th node. The unit

vectors and e2 can be obtained from the relations

i(-(6,x 61>/16 i)l

ii: tilx ('")

where i2 is the unit vector along the (global) x2-axis. If we assume

that the angles eI and e are very small, then we can write

51

t el+t ie (17)

Substituting Eq. (17) into Eq. (IS), we obtain

ui J=l(18a)

or

{u}- [Tl{a} (IBb)

where {u} is the column of three displacements at a point, {a} is the

column of 5n (five per node) displacements: u , e , e , j = 1,2,...,n; i

= 1,2,3, and IT] is the transformation matrix defined by Eq. (18a).

Thus for each time step one can find the normal vectors from Eq. (16)

and (17), and the incremental displacements at each point from Eq. (18)

once the five generalized displacements at each node are known.

Element Stiffness Matrix

The strain-displacement equations (4) can be expressed in the

operator form

{e} = {Al{u o} (Ig)

where {e} = {e11 e22 e33 2e12 2e13 2e23}T, [AI is a function

of tUoi,j, and {Uo} is the vector of the components of the displacement

gradient

{Uo} = {Ul, l Ul, 2 Ul, 3 u2, l u2, 2 u2, 3 u3, l u3, 2 u3,3 }T (20)

The vectors {Uo} and {e} are related to the displacement increments by

{Uo} = [N]{u} = [N)[TI{A} (21)

{e} = [A][N][T]{A} _ [B]{_} (22)

where [N) is the operator of differentials.

52

Substitution of Eq. (22) into Eq. (8) yields

J" Po{TIt{_}dVo + (t[KL] + t[KNLI){A } = t+At{R} - t+At{F} (23)Vo

where t[KL], t[KNL], {R}, and {F} are the linear and nonlinear stiffness

matrices, force vector, and unbalanced force vectors:

tIKL] = J"V

0

t[BITIc] tIB]dV o t t[ t[KNL] = [ B]T[s] [B]dV o

Vo

{F} = ; t[BIT{s}dV o (24)

Vo

Here [S] and [S} denote the matrix and vector, respectively, of the

second Piola-Kirchhoff stress.

Since we are dealing with laminated composite structures, the

important thing is how to perform the integration through the

thickness. One way is to pick Gaussian points through the thickness

direction. This increases the computational time as the number of

layers is increased, because the integration should be performed

separately for each layer. An alternative way is to perform explicit

integration through the thickness and reduce the problem to a two

dimensional one. The Jacobian matrix, in general, is a function

of {i' E2' and ;. The terms in ; to the first power may be neglected,

provided the thickness to curvature ratios are small. This

approximation implies that derivative of xI with respect

to _I' 42' and ; are substantially the same at either end of a mid-

surface-normal llne. Thus the Jacobian [J] becomes independent of ; and

explicit integration can be employed. If ; terms are retained in [J},

53

Gaussian points through the thickness should be added. In the present

study, it is assumedthat the Jacobian is independent of {.

Time Inteqration

The Newmark integration scheme Is used to convert the ordinary

differential equations in time, Eq. (23), to algebraic equations. In

the Newmark scheme, displacements and accelerations are approximated by

t+At{A} : t{A} + At2{_} + [(½_ B)t{_} + Bt+at{A}](_t)2

t+_t{_} = t{_} + [(l - y)t{_} + Yt+At{_}]At (25)

where {A} is the generalized displacement vector of any point

and B and y are the dimensionless parameters of the approximation. For

the constant average acceleration case, we have B = and y = _, and for

i (see [II])the linear acceleration method B = and y = _

Substituting Eq. (25) into Eq. (23), and some algebraic

manipulation leads to

(aot[Ml + t[K]){A(k)} = t+At{R } _ t+At{F(k-l) } + a3{P 4}

where

1 t _t{+ a2It{Pi } - _( {P2} P3})I (26)

1 1 1ao = a2 =_ a3 - l and

B"_t'2t} ' BAt ' =)-B '

[M) : j" oo t[T)T tiT] dVoV0

54

{PI } = ]" _o t{_} [T]dVo

Vo

{P2}=fVo

oo t+at{a}(k-l)[T]dV o

{P3 } = [ Po t{a}IT]dVo

Vo

{P4} = F 10o t{_} [T]dVo (27)

Vo

This completes the finite-element formulation of the 3-D degenerated

element.

DISCUSSION OF THE NUMERICAL RESULTS

The results to be discussed are grouped into three major

categories: (1) static bending, (2) natural vibration, and (3)

transient response. All results, except for the vibrations, are

presented in a graphical form. All of the results presented here were

obtained on an IBM 370/3081 computer with double precision arithmatic.

Static Analysis

Here we present a discussion off our example problems, all

involving shell structures.

I. Cylindrical Shell Subjected to Radial Pressure Consider a

circular cylindrical panel of the type shown in Fig. 3. The shell is

clamped along all four edges and subjected to uniform radial inward

pressure. The loading is nonconservative, that is, the direction of the

applied load is normal to the cylindrical surface at any time during the

deformation. The geometric and material properties are

55

X3=z

2=Y

I=X

c \free edge

supportedby a rigid

diaphram: u =

Figure 3 Geometry of the cylindrical shell used in Problem l

of the static analysis.

56

R = 2540 mm, a = b = 254 mm, h = 3.+75 mm,

e = O.l rad, E = 3.10275 kN/mm 2, v = 0.3

Due to the symmetry of the geometry and deformation, only one quarter of

the panel is analyzed. A load step of 0.5 KN/m 2 was used in order to

get a close representation of the deformation path. Fig. 4 contains the

plot of central deflection versus the pressure. The solution agrees

very closely with that obtained by Dhatt [13].

2. Orthotropic Cylinder Subjected to Internal Pressure Consider a

clamped orthotropic (E2 = 20 x lO6 psi, EI/E 2 = 3.75, G12/E 2 =

0.625, v = 0.25) cylinder of radius R = 20" and length 20", and

subjected to internal pressure, Po = 6.41/_ psi. A mesh of 2x2 nine-

node elements is used to analyze the problem. The linear center

deflections obtained by the 2-D and 3-D elements are 0.0003764 in., and

0.0003739 in., respectively. These values compare favorably with

0.000366 in. of Rao [14] and 0.000367 in. of Timoshenko's analytical

solution [15]. The latter two solutions are based on the classical

shell theory.

In the large-deflection analysis the present results are compared

with those of ReFerence 5. A value of 2.5 ksi is used for the load

step. Figure 5 contains a comparison of the present deflection with

that of Reference 5, which used a 3-D degenerated element based on the

updated Lagrangian approach. The agreement is very good.

3. Nine-Layer Cross-Ply (0°/90°/0°/...) Spherical Shell Subjected

to Uniform Loadin 9 Consider a spherical shell laminated of nine layers

of graphite-epoxy material (EI/E 2 = 40, G12/E 2 = 0.6, G13 = G12 =

G23' v12 =.25), subjected to uniformly distributed loading, and simply

supported on all its edges (i.e., transverse deflection and tangential

57

3.0

A

s2.5

E

2.0

O

S,.

e-

L

1.0

0.5

O°

o 3-D Element

o 2-D Element

Reference [13]

0. 2. 4. 6. 8. I0.

Center deflection, w (in ram)

12.

Figure 4 Load-deflection curve for the clamped cylindrical shell

58

v

%-

%-C:.

c-%-_Jww_

I0.0

8.0

6.0

4.0

Reference [ 5]

3-D Element

Linear solution

2.0

0.0

Figure

0.25 0 0 I l.25

Vertical center deflection (in.)

5 Center transverse deflection versus

internal pressure

59

rotations are zero). A comparison of the load-defl'ection curves

obtained by the present elements with those obtained by Noor {4] is

presented (for the parameters h/a = O.Ol and R/a = lO) in Fig. 6. The

results agree very well with each other, the present 2-D results being

closer to Noor's solution. This is expected because Noor's element is

based on a shell theory.

4. Two-Layer Cross-Ply and Angle-Ply (45°/-45 °) Shells Under

Uniform Loadinq The geometry of the cylindrical shell used here is the

same as that shown in Fig. 3. The shell is assumed to be simply

supported on all edges. The material properties of individual lamina

are the same as those used in Problem 3. A mesh of 2x2 nine-node

elements in a quarter shell is used to model the problem. The results

of the analysis are presented in the form of load-deflection curves in

Fig. 7. From the results, one can conclude that the angle-ply shell is

more stiffer than the cross-ply shell.

The geometry and boundary conditions used for the spherical shells

are the same as those used in Problem 3. The geometric parameters used

are: R/a = lO, a/h = lO0. The load-deflection curves for the cross-ply

and angle-ply shells are shown in Fig. 8. From the plot it is apparent

that, for the load range considered, the angle-ply shell, being stiffer,

does not exhibit much geometric nonlinearity. The load-deflection curve

of the cross-ply shell exhibits varying degree of nonlinearity with the

load. For load values between lO0 and 150, the shell becomes relatively

more flexible.

Natural Vibration of Cantilevered Twisted Plates

Here we discuss the results obtained for natural frequencies of

various twisted plates. This analysis was motivated by their relevance

60

c-O.r--

w-_P

c-

C.J

4.0

3.0

2.0

l.O

s f

/z' -- 2-D and 3-D Elements

I I " 1 " i |"

0 2 4 6 8 lO

Load, _ =(pa4/E2 h4)

Figure 6 Deflection versus load parameter for a nine-

layere cross-ply (0°/90°/0°/... ) spherical

shell

61

t--

O

U

Z.,:#

C

U

N

%t-o

E,m..

oz

2.5-

2.0-

1.5-

1.0

0.5

0.0!

0 5() 1O0

Figure

o 2-D Element

o 3-O Element[0°190°]

, 3-D Element, [450/-45 °]

! i ! i i !

150 200 250 300 350 400 450

Load parameter, _ : (pa4/E2h4)

Def]ection versus the load parameter for two-layer

composite cylindrical shell

62

AP CP

0.24,3.0GO,F--

U

,u 0.20,2 5m-,- •u,.,

S,-

0.16,2.0(.J

_J

%8 0 2,15

e-

QJE

c-

O 0.08,1.0Z

0.04 ,.5

0.0

AP = Angle-ply, CP = Cross-ply

o 2-D and 3-D Elements

m 3-D Element

o _45o]

0 20 40

Figure 8

i T T ,,

60 80 100 120 140 160 180 200 220 240

Load parameter, _:(pa4/E2 h4)

NondimensionaIized deflection versus the loadfor laminated shells

63

to natural vibrations of turbine blades. Consider an isotropic

cylindrical panel with a twist angle B at the Free end. Table l

contains the natural frequencies of a square plate for various values of

the twist angle e and ratios of side to thickness. A 2x2 mesh and 4x4

mesh of g-node elements are employed to study the convergence trend.

The results of the refined mesh are included in the parentheses. The

results obtained by using the 4x4 mesh are lower than those predicted by

the 2x2 mesh, showing the convergence. The results agree with many

others published in a recent NASA report. Table 2 contains natural

frequencies of twisted plates for the aspect ratio of 3.

Transient Analysis

I. Spherical Cap Under Axis_mmetric Pressure Loadinq Consider a

spherical cap, clamped on the boundary and subjected to axisymmetric

pressure loading, Po" The geometric and material properties are

R = 22.27 in., h = 0.41 in., E = I0.5 x lO6 psi, _ = 0.3,

= 0.095 Ib/in 3, e = 26.67 °, Po = lO0 psi, _t = lO-5 sec.

This problem has been analyzed by Stricklin, et al. [161 using an

axisymmetric shell element. In the present study the spherical cap is

discretized into five nine-node 2-D and 3-D elements. Figure g contains

the plot of center deflection versus time. The present solutions

obtained using the 3-D and 2-D elements are in excellent agreement in

most places with that of Stricklin et al [16]. The difference between

the solutions is mostly in the regions of local minimum and maximum.

2. Two-Layer Cross-Ply Plate Under Uniform Load A cylindrical

shell with a = b = 5", R = lO", h = O.l" is simply-supported on the four

edges, is analyzed. The shell is laminated by 2 layers (o°/go °) and

exerted by a uniform step load P = _a4P = 50. Figure lO contains a plot

E2h4

64

Table Natural Frequencies

--: wa2/_-h-/D D -

of Twisted Square

Eh 3_J

2 'l2(l-v )

Plates

0.3

ah

Twist

Angle

M)de

1 2 3 4 5 6

2O

0° * 3.4556 8.4110

(3.4583) (8.3353)

15° 3.4359 I0.2920

30° 3.3790 13.7014

(3.3694) (14.2222)

4.5° 3.2908 18.1009

60 ° 6.1800 17.8319

22.0999

(21.0238)

21.5199

19.9840

(18.9795)

15.9097

15.5635

28.2089

(26,7465)

27.2054

25,0943

(26.8104)

23.5680

24.1842

31.9740

(30.1 454)

32.7430

34.3341

(34.4591)

35.5332

36.1466

55.1625(52.0784)

44.5375

45.8987

(45.7547)

45.7013

44.9152

0

15 °

30 °

45 °

60 °

* 3.33916 7.3948

**(3.3390) (7,3559)

10.8083

(1 0.883)

3.31713 7.4816 10.8053

(3.3170) (7.4504 (I0.774)

3.2538 7.7593 10.5248(3.2538) (7.7089) (1 0.478)

3.1570 8.1435

(3.1 569) (8.0728)

3.0370 8.5855(3.0366) (8.4814)

I0.1270

(10.062)

9.67198

(8.5911)

18.4930

(17.757)

18.4043

(17.771)

18.4091

(17.795)

18.3843(17.79)

18.3089

(17.730)

23.7907

(22.769)

23.6767

(22.694)

23.3734(22.471)

22.9126

(22,117)

22.3670

(21.684)

26.0552

(24.125)

24.9474

(24.083)

24.6116

(23.943)

24.0566

(23,651)

23.3533(23.160)

* 2x2,

*'3x3,

t 4X4,

9-node mesh9-node mesh9-node mesh

65

Table Natural Frequencies of Twisted Rectangular Plates

(b/a = 3, 3x3 mesh of nine-node elements)

_ = _b2V-GTTU , D = Eh3 , _ = 0.3

12(I-v) 2

a Twi sth" Angle

Mode

1 2 3 4 5 5 7

20

o

15°

30°

45 °

60°

-- "(_°

15°

30°

45°

60°

3.4150

3.4009

3.3598

3.2956

3.2136

3.3908

3.3161

3.3336

3.2674

3.1833

20.8772 21.6190 65.9706 66.2590 127.256

20.8798 22.1118 21.5032 68.0938 69.3258 130.284

19.4048 25.3743 60.2183 73.5180 77.4493 138.176

17.5289 29.8404 58.2600 80.9488 88.5245 148.8975

15.7431 34.8827 55.8921 8g.2028 I00.7760 155.070

15.551 19.124 21,065 59.924 61.949

15.192 19.231 21.572 60.088 60.830

14.379 19.549 22.811 60.576 58.472

13.449 20.060 24.404 51.360 55.874

12.548 20.741 26.139 62.416 53.381

66

-l .0-

20 40 BO ,_00:Time x 103 sec,

I, /

i ip

i ,e

120

i

Figure 9 Center transverse displacement versus time

for a spherical cap under axisymmetric dynamic

loading ( load : lO0 psi.)

67

0.7 -- 3-D Element

.--2-O Element

O.l

0.0

0 l

Figure I0

2"" 3

Time, txlO4

4 5

?

Center deflection versus time for two-layer

cross-ply cylindrical shell subjected touniform step load

68

of the center deflection versus time for 2-D and 3-D elements. The time

step used is _t = O.l x lO-4 sec. The solutions obtained using the two

elements are in good agreement.

3. Two-Layer An_le-Ply (45°/-45 °) Spherical Shell Under Uniform

Loadinq Consider a spherical shell with a = b = lO", R = 20" and h =

O.l", simply supported at four edges and is exerted by a uniform step

load. The shell consists of two layers, (45°/-45°). Figure II contains

the plot of center deflection versus time for P = 50 and P = 500 with

time step 0.2 x lO-5 sec. For the small load the curve is relatively

smooth compared to that of the larger load. This is due to the fact

that the geometric nonlinearity exhibited at P = SO is smaller compared

to that at P = 500.

CONCLUSIONS

The present 3-D degenerated element has computational simplicity

over a fully three-dimensional element, such as those developed in [17J,

and the element accounts for full geometric nonlinearities in contrast

to 2-D elements based on shell theories. As demonstrated via numerical

examples, the deflections obtained by the 2-D shell element deviate from

those obtained by the 3-D element for deep shells. Further, the 3-D

element can be used to model general shells that are not necessarily

doubly-curved. For example, the vibration of twisted plates cannot be

studied using the 2-D shell element discussed in [12]. Of course, the

3-D degenerated element is computationally more demanding than the 2-D

shell theory element for a given problem. In summary, the present 3-D

element is an efficient element for the analysis of laminated composite

plates and shells undergoing large displacements and transient motion.

The 3-D element presented herein can be modified to include thermal

stress analysis capability and material nonlinearities. While the

69

0.7

0.6

0.5

_= 0.4

0

.p,.

U

•"- 0.3

C

_ 0.2

L

I

--wxlO 3 for load parameter, _ = 50

wxlO 2 for load parameter, P = 500

O.l

0.0

Figure II

4.0 6.0

Time, txlO5

8.0 I0.0

Center deflection versus time for two-layerangle-ply [45:/-45 °] spherical shell under

uniformly distributed step loading.

70

inclusion of thermal stresses is a simple exercise, the inclusion of

nonlinear material effects is a difficult task (see (18-20]). An

acceptable material model should be a generalization of Ramberg-Osgood

relation to an anisotropic medium. Another area that requires further

study is the inclusion of damping effects, which are more significant

than the shear deformation effects.

71

o

o

o

o

o

m

o

o

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15.

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20.

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D. R. J. Owen and J. A. Figueiras, "Anisotropic elast-plasticfinite element analysis of thick and thin plates and shells," Int.

J. Numer. Meth. Enqnq., 19(4), 541-566 (1983).

T. Kuppusamy, A Nanda, and J. N. Reddy, "Three-dimensional analysis

of composite plates with material nonlinearity," Report No. VPI-E-

83.34, Virginia Polytechnic Institute and State University,

Blacksburg, VA, 1983.

73

PART3

NONLINEARMATERIALMODELSFORCOMPOSITEPLATES AND SHELLS

K. Chandrashekhara and J. N. Reddy

Department of Engineering Science and Mechanics

SUMMARY

Nonlinear material models for laminated structures are described

and their incorporation In the finite-element formulation of laminated

plates and shells is presented. Numerical results for several sample

problems of plates and shells are presented and validated by comparison

with those available In the literature.

INTRODUCTION

Composite materials are known to exhibit significant non-

linearities in stress-strain behaviour even at low strains. Most of the

currently used matrix materials in composites have high strain

capabilities and the investigation of the bending of composite shells

undergoing large deformation, yielding is apt to occur and its effect

must be accounted for in the analysis. The nonlinearity is not

isotropic but varies wlth direction, as do the elastic properties.

Models for such elastic-plastic behavior of orthotropic and anisotropic

materials are not well developed.

The total stress-strain laws are mathematically more convenient

than incremental laws but are physically not sound. The criterion

approximately describing the yielding of isotropic material Is that of

von-Mises. The simplest yield criterion for anisotropic material is

therefore one which reduces to von-Mises law when the anisotropy is

vanishingly small. Hill's yield criteria assumes relatively simple ease

of orthotropic anisotropy, that is, there are three mutually orthogonal

74

planes of symmetry at every point and the intersection of these planes

are considered as the principal axes of anisotropy. Fiber reinforced

composite structures almost invariably possess this kind of symmetry.

In the present study a nonlinear material model is developed for

composite plates and shells, and numerical results for bending are

presented using the finite element method as exact solutions are not

tractable for elastic-plastic problems involving complex geometries.

MATERIAL MODEL

In the present model, Hill's anisotropic yield criteria for

elsatic-perfectly plastic material is used. Hill's [1] yield function

is,

f(alj) = F(a 2 - o3)2 + G(o 3 - oi)2 + H(o I - o2)2

where F, G, H, L, M, N are parameters characteristic of the current

state of anisotropy given by,

I I I

1 1 1

; 2L = 1R2

ZM = 1

S2

1 1 1 =1

and X, Y, Z are the tensile yield stresses in the principal direction of

anisotropy and R, S, T, are the yield stresses in shear with respect to

the principal axes of anisotropy.

It should be noted that Hill's criteria is based on the assumption

that the superposition of a hydrostatic stress does not influence

yielding and there is no Bauschinger effect. Also, the yield criterion

75

has this form when the principal axes of anisotropy are the axes of

references.

For a plane stress state in the I-2 plane with transverse shear,

equation (1) reduces to:

f = (G+H)o_+ (F+H)a_- 2HOlO 2

+ ZLo_3 + 2M0_3 + 2N0_2 = l (2)

For an Isotropic material;

X = Y = Z = %,

the yield stress in uniaxial tension and according to the von-Mises

yield criteria [2]

Therefore, F = G = H =

°oR_-S_-T=M

becomes,

1and 2L = 2M = 2N -

200

and equation (2)

°0

which is the familiar von-Mises yield criteria.

If the principal axes of anisotropy 1,2 do not coincide withthe

reference axes x, y, but are rotated by an angle e, then the stresses in

equation (2) are obtained using the transformation as:

01 = ox cos2e + Oy sin2e + Oxy sine cose

o2 = ox sin2e + Oy cos2e - axy sine cose

023 = -Oxz sine + Oyz cose

o13 = axz cose + ay z stne

o12 = -2a x sine cose + 2ay sine cose + axy(COSZe - stn2e)

Elastic-Plastic Constitutive Equations

In the incremental theory of plasticity, the total strain increment

is the sum of the elastic and plastic components

76

dc = d¢e + d=p "' (3)

The elastic strain increment is related to the stress increment by

Hooke's law as,

dce = [oe]-ldo (4)

where [De ] is the elastic modulus matrix which for orthotropic material

takes the ?orm,

[D e ] =

m

E1 v12E2 0 0 0

1-v12v21 l-v12v21

v12E 2 E20 0 0

l-v12v21 l-v12v21

0 0 G23 0 0

0 0 0 G13 0

0 0 0 G12_0

(s)

The normality rule for an associated plastic flow is,

dep = dx _f@o

where dx is the positive proportionality constant, evaluated using the

condition that during the plastic deformation, the stresses remain on

the yield surface so that,

df = =--J-'do = 0_o

The stress-strain relation in the plastic range is given by [3],

do = [DeP]d¢

where

_f T De[D el [-_o} {T'_o} [ ][Dep] = [De] - (G)

{_a-_fo}T[De]{_a_fo}

77

Hencethe modification called for in the elastic-plastic analysis

would be solely the replacement of the elasticity matrix [De] by the

elastic-piastic matrix [Dep] for the yielded elements at the successive

stages of calculation. It should be noted that the [Dep] matrix is

populated and accordingly the transformation of the stress-strain

relation from the material axes, ({a} = [DeP]12{¢}), to the shell

coordinate axes, ({a} = [DeP]xy{¢}), will be modified as shown in

Appendix I.

FINITE ELEMENT FORMULATION

Consider a laminated shell constructed of a finite number of

uniform thickness orthotropic layers, oriented arbitrarily with respect

to the shell coordinates (_1,{2,{). The orthogonal curvilinear

coordinate system ({1,{2,¢) is chosen such that {1- and {2- curves are

lines of curvature on the midsurface {=0, and {-curves are straight

lines perpendicular to the surface {=0.

For the small displacement Sanders shell theory which accounts for

transverse shear deformation, the strain displacement relations are

given by ISI,

where

¢I = ¢i + {(i

au I u3 a®1

au2 u3 _¢2

o aUl au2 a_1 a_2 au2 aUl

•6" + ; + Coo au3 u2

¢4 = ¢2 + _x2 R2

78

o au3 Ul

_5 = _1 + ax 1 R1

Co =½ (R_- R_) , dx i " _id¢i(l:l 9 2 )

Here Ri (i = 1,2) are the principal radii of curvature, ui are the

displacements of the reference surface along {i(¢3 = {) axes, ¢i and _2

are the rotations of the transverse normals about the ¢2 and {1-axes

respectively.

The stress-strain relations, transformed to the shell coordinates,

are of the form

{o}- {QI{,}

where QkiJ are the material properties of kth-layer (see Appendix I).

The principle of virtual work for the present problem is given by

._

i=I

+ o_k) 6t5 . q6u3}Q1_zdEid¢2id; (7a)

= ; [N16c _ + N26_ _ + N66¢ _ + MI_( I + M26_ 2 + .6_( 6

+ N46¢ _ + N56¢ _ - q6u3}=1=2dE1d¢ 2 (Tb)

where q is the distributed transverse load, Ni and Mi are the stresses

and moment resultants.

L ;k

(Ni,Mi) = _ _ ai(l,;)d;

k=1 ;k-1

(i = 1,2,6,4,5)

Here ({k.l,¢k) are the ;-coordinates of the kth layer, and L is the

total number of layers in the laminated shell.

79

and setting the coefficients of _ui to zero separately.

t

It should be noted that the equations of equilibrium can be derived

from Eq. (Tb) by integrating the displacement gradients in ¢_ by parts

We obtain

aN1 a N5

+_ (N6+CoM6)+_ --0a aN2 N4

_x_(%- co%)+_x-3 +_ "oaN5 aN4 N1 N2

q).o_M1 aM6

_'_"1+ _'_2 - N5 " 0

_M6 _M2

a-'_"l+ _-_'2 - N4 = 0

0The resultants (Ni, Mi) are related to (¢i,(i) by,

0

Ni = Aijcj + Bzp_P i,j = 1,2,6,4,5

0

M_ = Bzj¢ j + Dzp_ p z,p = 1,2,6 (with z=i for i = 1,2,6)

(8)

Here Aij, Bij and Dij denote the extensional, flexural-extensional

coupling, and flexural stiffnesses of the laminate:

L _;k

(Aij,Bij,Dij) = z ]"k=l _k-1

(k) (I,_,¢2)d¢Qij

In the unabridged notation equation (8) takes the form:

(9)

80

N1

N2

N6

N4i

N5

MI

M2

M6 .

,z

--All AZ2 A16 _A14A_lS BZZ BZ2 B16-

A12 A22 A26 A_24 A_25 B12 B22 B26

A16 A26 A66 A46 A56 B16 B26 B66

A14 A24 -A46 A44 A45 _B14 B24 _B46

AI5 A25 A56 A45 A55 BI5 1325 B56

BII B12 B16 B-14 B15 D11 D12 D16

B12 B22 B26 B24 B25 Dll D22 D26

B16 B26 B66 B46 B56 D16 026 066

0

¢1

0

¢2

0

¢6

0

¢4. (10)

0¢5

_2

The underscored coefficlents are due to material nonlinear stress-strain

relationship. It should be noted that the coefficients A44, A45 and ASS

defined in equation (g) has to be corrected for the parabolic variation

of the transverse shear stress• as

L {k

.Fk rL'Zn(k)= _i_44 ,(A44•A45•A55) k1:l -1

(k)•(_Q(k))d; (ll)klk2Q45 55

where k are the shear correction factor.

A typical finite element is a doubly-curved shell element whose

projection is an isoparametric rectangular element. Over the typical

shell _(e), the displacements (u1,u2,u3,¢1,¢ 2) are interpolated by

expressions of the form,

N

u_$j(Xl,X 2)ui = j=l

N4

= r ¢J@j(Xl,X2),-'_ij=l

, i = I•2,3

• i =1,2

(12)

where _j are the interpolation functions, and u_ and ®_ are the nodal

81

values of ui and ei, respectively. For a nine node quadratic element

the element stiffness matrix is of order 45x45.

Substitution of equation (12) into the virtual work principle, Eq.

(7b) yields an element equation of the form

[K]{a}- {F} (13)

where {a} - {{Ul}, {u2}, {u3}, {el}, {e2}}T, [K] is the element

stiffness and {F} is the force vector. In the interest of brevity, the

coefficients of stiffness matrices are included in Appendix If.

It should be noted that the underscored coefficients in Eq. (lO)

are also redefined like the shear coefficients in Eq. (ll) and reduced

integration is performed for the terms arising in the element stiffness

matrices due to the presence of these coefficients to avoid the so-

called locking effect.

NUMERICAL RESULTS

The Parameters of Anisotrop_

When considering the modeling of a material system, one must always

survey the availability of material property data. In the present

theory, to describe fully the state of anisotropy, the six independent

yield stresses in Hill's criteria are needed to be known from uniaxial

tests. For numerical results, two typical composite materials namely,

boron/epoxy and graphite/epoxy are considered with the following

material constants:

Boron/Epoxy

E1 - 30.0 x lO6 psi , E2 = 3.2 x lO6 psi

G12 = 1.05 x lO6 psi , _12 = 0.21 , G23 = G13 = G12

X = 195 x 103psi ; Y = Z = 12.5 x lO3 psi

R = S = T = 18.0 x lO3 psi

82

Graphite/Epoxy

E1 = 18.88 x 106 psi ; E2 = 1.376 x lO6 psi

G12 = 0.688 x lO6 psi ; v12 = 0.343 ; G23 - G13 - G12

X - 222.7 x 103 psi ; Y = Z - 6.35 x 103 psi

R = S - T = 9.92 x 103 psi

Solution Procedure

The solution of the elastic plastic problem ts reached by an

incremental and iterative procedure. The direct iteration technique is

followed in the present analysis.

For each load increment, the system of equations are established by

assemblying the element matrices and the displacement {a} is obtained

from Eq.(13). Consequently, the state of stress and the value

of f(aij ) are calculated for each element. If f < O, then the process

is elastic and the material matrix is obtained from equation (S). If f

> O, then the total stresses are readjusted so as to make f = 0 and the

elastic-plastic matrix is calculated from Eq. (6). Once the convergence

is achieved, the next load increment is applied and the iteration

procedure is repeated.

If the application of a small load increment causes very large

deflection, the calculation is stopped and the limit load is considered

to be found.

Sample Problems

The present elastic-perfectly plastic formulation is applied to a

variety of bending problems using 2xZ mesh of a nine noded quadratic

element. The shear correction factors k_ = k_ were taken to be 516.L

83

All computations were made using an IBM 3081 processor with double

precision arithmetic.

The results of the sample problems are presented and compared, if

possible, with the existing solutions to evaluate the present

formulation.

I. Cylindrical Shell Roof A cylindrical shell subjected to

uniform vertical loading is considered. Due to symmetry, only a

quadrant of the shell was analyzed. The geometry and modeling of the

shell roof are shown in Fig. I. The material behaviour is studied with

the properties:

E1 = E2 = 2.1 x lO4 MN/m 2 ; v = 0.0;

G12 = 1.05 x lO4 MN/m 2 ; G23 = G13 = G12

X = Y = Z = 4.1Mn/m 2 ; R = S = T = 2.367 MN/m 2

The results obtained for the vertical displacement at the central

point of the free edge A versus loading was shown in Fig. I. The

solution obtained compares well with those reported in Ref. [6]. The

apparent discrepancy can possibly be due to a different boundary

condition on the curved edges and the type of material model used.

2. Simply-Supported Square Plate A uniformly loaded simply

supported square plate was studied in the second example. The geometry

of the plate is shown in Fig. 2. The following material properties were

X : Y = Z : 144,000 psi

considered:

E1 = E2 - I0 x 106 psi ; v = 0.3

G12 : 3.846 x lO6 psi ; G23 = G13 : G12

; R - S : T = 83,138.4 psi

84

3.0 "

2.5

2.0

_(kNlm 2 )

1.5

0 = O"o y

Present solution

Reference [6]

= 4.I MN/m2

ff

l.O

0.5

0.0

0

Y

xL Free edge

o L

\IxTR L:716m' / h:O.O76m

l I l I

•Ol .02 .03 .04 .05

Vertical deflection at point A (in meters)

Figure I. Load-deflection curves for a cylindrical panelunder uniform transverse load

85

0.8

0.7

0.6

0.52

Po__ a6Mo0 4

C).3

0.2

0.!

0.0

Present solution

.... Reference [7] /-_

//

o° = 14,400 psi_

/ a:2in.

h = 0.05 In)Mo= (_oh2/4

X

, l ! ! l

0.0 .02 .03 .04 .05 .06

Center deflection, wD/(_Moa2)

Figure 2, Load-deflection curves for a simply-supportedsquare plate under uniform transverse load

86

A non-dimensionalized plot of the centre displacement of the plate

versus the load are shown in Fig. 2. The results are compared with

those presented in Ref. {7].

3. Two Layer Cross-Ply [0/90] and Angle-ply {-45/45] Simply

Supported Spherical Shells Figure 3 contains the results for the cross-

ply shell made of two typical materials, namely, boron/epoxy and

graphite/epoxy under uniform load. For a given load, the shell made of

graphite/epoxy deflects more than the shell made of boron/epoxy which is

stiffer, but experiences small degree of nonlinearity.

Figure 4 shows nonlinearity exhibited by the graphite/epoxy cross-

ply and angle-ply shells under uniform load. Clearly, the angle-ply

shows greater displacement and also nonlinearity than the cross-ply for

the same load.

Figure 5 shows the material behaviour for the boron/epoxy cross-ply

shell under concentrated load.

4. Clamped Cylindrical Cross_Pl X (0/90) Shell Under Uniform Load

The geometry of the shell is shown in Fig. 6. The shell is made of

grpahite/epoxy and the plot of displacement versus load are shown in

Fig. 6.

CONCLUSIONS

A finite element model based on Sander's shell theory, accounting

for the transverse shear strains is used for the elastic-plastic

analysis of lamianted composite shells. The parameters of anisotropy

reflect the plastic material response by correcting the stress

components in the Hill's yield function. Numerical results are

presented for isotropic and laminated shell of cylindrical and spherical

geometry to demonstrate the validity and efficiency of the present

87

I00

8O

Pc)(psi).

6O

4O

2O

0

-- Boron-epoxy I (0°/90°)

--- Graphite-epoxyJ

/

/

//

p/

/

Linear /Linear /

/ x

a = b = 50 in._

R : lO00 in.h = 1 in.

4 6 8 I0

Center deflection, w (in.)

Figure 3. Load-deflection curves for a simply supported

spherical shell under uniform transverse load

88

100

80

60

Po(psi)

4.0

2O

0

u Graphite-epoxy (0°/90 °)

--- Graphite-epoxy (-45°/45 °)

/

I I I I I

0 2 4 6 8 I0

Center deflection, w (in.)

Figure 4. Load-deflection curves for a simply supportedspherical shell (see Figure 3 for the geometry)

89

3.2_

Linear

2._

1.6

PoXlO-61b

0.8

0.0l D ! | i

0 2.5 5.0 7.5 lO.O 12.5

Center deflection, w (in.)

Figure 5. Load-deflection curves for a simply supported spherical shell

under point load at the center (see Figure 3 for the qeometry)

90

\

£...j

OJ,.

, :Q.) ._

U'_ P"

II

•_ "qP-

C:C

_.J tl

¢J

0 _

_.,- _'_

•,- _ E

•._ c-

O 0 e--

i,

91

approach. For the is,tropic case, the present results are in good

agreement with those available in the literature.

•

•

o

4.

•

•

.

REFERENCES

R. Hill, "A theory of yielding and plastic flow of anlsotropic

metals," Proc. Roy. S,c. Lond, Ser Alg3, 1948.

A. Mendelson, Plasticity: Theory and Application, The MacMillan

Company, New York, 1968.

G. C. Nayak and O. C. Zienkiewicz, "Elastic-plastic stress

analysis-a generalization for various constitutive relations,

including strain softening," Int. J. Num. Meth. Engn _. 5, I13-1351972.

J. N . Reddy, Energ_ and Variational Methods in Applied Mechanics,John Wiley, New York, 1984.

j. N. Reddy, "Bending of laminated anisotropic shells by a shear

deformable finite element," Fiber Science and Technology I_7,pp. g-24, 1982.

J. Backlund and H. Wennerstrom, "Finite element analysis of

elastic-plastic shells," Int. J. Num. Meth. Engng. 8, pp. 415-424,1974.

H. Armen, A. Pifko and H. S. Levine, "Finite element analysis of

structures in the plastic range, NASA contractor report, CR-1649,1971.

92

APPENDIX I ,,

Transformation of the stress-strain matrix in Equation (6)

Let the elastic matrix in the material axes (1,2) be [DeP]12 and in

the body axes be [DeP]xy

[DeP]12 = [C] =

-CII C12 C13 C14 CI_

% c24%

C33

sym

C34 C35

C44 C45

C55!

DeP =[Q]=[ ]xy

"Qll Q12 Q13 Q14 Ql;

Q22 Q23 Q24 Q25

Q33 Q34 Q35%

Q44 Q45

Q551

sym

then the transformation [4] is given as, (with m = cose, n = sine)

Qll = m4Cli + 2m2n2(C12 + 2C33) - 4mn(m2C13 + n2C23) + n4C22

Q12 = m2n2(C11 + C22 - 4C33) + 2mn(m2 - n2)(C13 - C23) + (m4 + n4)C12

Q13 = m2(m2 - 3n2)C13 + mn[m2C11 - n2C22 - (m2 - n2)(C12 + 2C66)]

+ n2(m 2 _ n2)C26

Q14 = m3C14 - mn[(2C34 " C15)m - (C24 - 2C35 )n] + C25 n3

Q15 = m3C15 - mn[(C14 + 2C35 )m - (C25 + 2C34)n] " C24n3

Q22 = n4C11 + 2m2n2(C12 + 2C33) + 4mn(m2C23 + n2C13) + m4C22

g3

Q23= m2(m2- 3n2)C23 + mn[n2C11 - m2C22 + (m2 - n2)(C12 + 2C33)I

+ n2(3m ? - n2)C13

Q24 " m3C24 + mn[(C2s + 2C34)m + (C14 + 2C3s )n] + CIs n3

Q25 " m3C25 - mn[(C24 - 2C35 )m - (C1S - 2C34)nl " C14n3

m2n 2 2mn(m 2 n2)(C22 C33) (m? n2)2C33Q33 " (Cll + C22 - 2C12) - - - + -

Q34 = (mC34 + nC3s )(m2 - n2) + m2n(C14 - C24) + mn2(CIs - C25)

Q35 = (mC35 - nC34)(m2 - n2) + m2n(CIs - C2s) + mn2(C24 - C14)

Q44 = m2C44 + 2mnC45 + n2C55

Q45 = (m2 - n2)C45 - mn(C44 - C55)

QSS : m2C55 - 2mnC45 + n2C44

The underscored terms are due to material nonlinearity for an

orthotropic material. Also note that the constitutive matrix is no

longer orthotropic.

94

Stiffness Coefficients

APPENDIX II

IK11] = AIIISII)+ A16([$12] + [$21]) + A66[$22]

+ Co(B16[$12] + [$21]) + 2B66[S 22] + CID66IS22] ) +

1 S01- _11(A15({sIO]+{ I)+A58(IS201+{sO2])-C°B56(IS20]+[S02))

IK12l = A12[$12 ) + A16[$11) + A26[$22] + A66[$21]

A4_5 [sOO I+ Co(B26[$22I - B16[$11I - COD66[$21]) + RIR 2

1 10]" _22 (A14[S + A46[S20] + C°B46[$201)

I

" R_I (A25{S02] + A56[S01] " C°B56[S01])

i R_ $10{KI3) = _i (Ali[$I0) + A16[$20]) + (A12[ ]

C {BI6 B26 IS2° S°2 (S01})°'R1 [$20] +R2-2 ] - R_ (A45[ ] + A55

A14[$12 ) IS11 S°_i°+ + A15( ] - ) + A461S22)

+ A56[$21] + COIB46[$22] + B56[$21] ) - A25 RIR 2

+ A26($20)) +

IK14] = B11[$11] + B16(IS12] + {S21]) + B661S22] +

S21] D66[S221 S°°Co(D16[ + ) - R_-A55[ l

+ A15[$10] + A56tS20] " R_" (B15[S01] + B56[S02])

+ COB56[$20]

[S12 S22 ] S21[K15] = B12 ] + B16[$11] + B26[ + B66[ ]

• 1 A45[sOO ]+ Co(D26[$22] + D66[$21] ) -

+ A14[$10] + A46[$20] - _ (B25[S02] + B56[S01] )

+ COB46[S20]

[K21] = [K12]T

[K22] A22[$22 ] $12] $11] 11]= + A26([ + [$21]) + A66[ - 2COB66[S

A44 [sOO]

_ Co(B26({$12 ] + [$21]) _ COD66[$11}) _ RT

1 $20 sO2] i01 10]- _22 (A24([ ] + [ ) + A46([S + [SOl] + c°[S )

- COB46[$10})

RL 1 (A22[S20 ] [slO])[K23] = (A12[$20] + A16[S10]) +]_"22 + A26

C (B16 B26_ [$I0] I S02 [sOl])- o RI + R2 ' - _22 (A44[ ] + A45

+ A24([$22 ] _ 1__ SooR_ [ ]) + A25[S21] + A46[$12]

$11 A14 soo] S12+ A56[ ] - R-_ [ - COB46{ ] - COB56[$11]

?

[K24] = B12[$21 ] + B26[$22] + B16[$11] + B66[$12]

I A45{Soo]- C°(D16[$11] + D66[S12]) - _22

I (B14[S01 ] [sO2])+ A25{$20] + A56[$I0] - _22 + B46

96

- CoB56{sIOl

[K25} = B22[$22] + B26(1S211 + 1S12]) + B661S II]

- Co(D261S121 + D661SII]) - )_2A44[S°°I

+ A24[$20] + A46 {SIO) - R_ (B24 [S02] + B46[S01])

- COB46[S 10]

[K31] = [KI31T

[K 32] = [K23} T

[K33] = A45[$12] + A55[S 11] + A44[$221 + A45[$211

,A12 A22)i (A11 A12, + "_2 'R1 + R2 )+ [S°°] ("_'i RI + R2 '

A14 A15

+ h'T-1(IS°21 + 1s2°I) +_'-i (IS°ll + [S10l)

A24 S02 S20 A2_..__5IS01 SI0+_(1 1 + { 1)+Rz ( l + [ l)

A24 S02] S20 ] A25 S01 SI0 )]+[ >

[K34] = A55[SI0] + A45[$20]

+ _.R1 + R2 "RI + R2

+ {A15 A25)[s0O l + B14[$21] + B46[$22],_RI + R2

+ B15{S11l + B56[S121

97

[K351 = A4515 I0] + A44[S 20]

,B_ I6++ _R1

_A14

+ _RT_ +

B26)[S 01] + ,B12 B22)[S02 lR2 VRT"1 + R2

A24_ soo S21-R-T-zJ[ ] + B24[S 22] + B46[ ]

+ B25[S 12] + B56[S 11]

{K41] = [K14] T

[K 42} = [K24] T

[K 43] = [K34] T

: SII S12 S21 S72 [S°° ][K44] 011[ ] + D16([ ] + [ ]) + 066[ ] + A55

+ B15([S I0] + [S01]) + B56([S 20] + [S02])

[K45] = 012[S 12] + 016[S 11] + 026[S 22] + D66[S 21] + A45[S°°]

+ B14[$I0] + B461S20] + B25[S 02] + B56[S 01]

[K51] = [KIS] T

[K 52] = [K25] T

[K53] = [K35] T

[K54] = [K45]T

[K55]= 022[S 22] + 026([S 12] + [S21]) + D66[S 11] + A44[S °°]

+ B24([S 20] + [S02]) + B46([S I0] + [S01])

98

where

QB__Sij

_eoo = _Qe _l_jdx1dx2

l_i ___dXldX2 , Si j

and the underscored terms are due to material nonlinearity.

99

FormApproveaREPORT DOCUMENTATION PAGE OMBNo0704.0188

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0attwr.',0 line maintaining me olta neeOeer, s_ oompWting Ira0 rme_mng me cet_a_ of m/onnabon. _ dominants regar_Jr_ thl burmm m'amale or any om_ m_oecl of th_¢oitectlo_ of r_l_to'L mcJuchr_l s_.s_ons _or mO_o_ _tts bucden, to Wluthtr_ He_fteft Ser_oB, D_lctorl_ tot m/oml_ol_ O_t_lm _ Relx>rtlt. 1215 JotlmlOn

Do,.ql Ht_y. Sure 1204. /ul_o_, VA 22202-4302, _ to Itw Offce of Mana_ _ BtX:_. Pl_erw_lt Roduc_ll0n Proleol (0704-0188), WIIM_O_, DC 20503

1. AGENCY USE ONLY (Leave b/ank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

January 1993

4. _I_.E AND SUBTITLE

Geometrically Nonlinear Analysis of Laminated Elastic Structures

¢ Aurmon(s)

J.N. Rcddy, K. Chandrashekhara, and W.C. Chad

7. PERFORMING ORGANIZ.ATION NAME(S) AND ADDRES$(ES)

9. SPONSORING/MONITORING AGENCY NAME$(SI AND ADDRESS(ES)

National Aeronautics and Space AdministrationLewis Research Center

Cleveland, Ohio 44135-3191

Final Contractor ReportS. FUNDING NUMBERS

WU-505-63-53

NAG3-208

L PERFORMING ORGANIZATION

REPORT NUMBER

E-7533

10. SPONSORING/MONWORING

AGENCY REPORT NUMBER

NASA CR-191055

11. SUPPLEMENTARYNOTES

Project Manager, C.C. Chamis, Structures Division, NASA Lewis Research Center (216) 433-3252,

12==. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category 24

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

This final technical report contains three parts: Part 1 deals with the 2-D shell theory and its element formulation and

applications. Part 2 deals with the 3-D degenerated element. These two parts constitute the two major tasks that were

completed under the grant. Another related topic that was initiated during the present investigation is the developmentof a nonlinear material model. This topic is briefly discussed in Part 3. To make each part self-contained, conclusions

and references are included in each part. in the interest of brevity, the discussions presented here are relatively brief. The

details and additional topics are described in the references cited.

.. SUJU,;eTTERMS

Shell theory; 2-D finite elements; 3-D finite element; Nonlinear material

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Unclassified

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Unclassified

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Unclassified

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A0520. LIMITATION OF ABSTRACT

Standard Form 298 (Rev 2-89)

Prescribed Dy ANSI St¢l Z39-18298-102

National Aeronautics and

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Chamis 49-8

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